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We initiate a study of the topological group PPQI(G,H) of pattern-preserving
quasi-isometries for G a hyperbolic Poincaré duality group and H an infinite
quasiconvex subgroup of infinite index in G. Suppose ∂G admits a visual
metric d with dimhaus < dimt + 2, where dimhaus is the Hausdorff dimension
and dimt is the topological dimension of (∂G,d). Equivalently suppose that
ACD(∂G) < dimt + 2, where ACD(∂G) denotes the Ahlfors regular conformal
dimension of ∂G.
- If Qu is a group of pattern-preserving uniform quasi-isometries
(or more generally any locally compact group of pattern-preserving
quasi-isometries) containing G, then G is of finite index in Qu.
- If instead, H is a codimension one filling subgroup, and Q is any group of
pattern-preserving quasi-isometries containing G, then G is of finite index
in Q. Moreover, if L is the limit set of H, L is the collection of translates
of L under G, and Q is any pattern-preserving group of homeomorphisms
of ∂G preserving L and containing G, then the index of G in Q is finite
(Topological Pattern Rigidity).
We find analogous results in the realm of relative hyperbolicity, regarding an equivariant
collection of horoballs as a symmetric pattern in the universal cover of a complete
finite volume noncompact manifold of pinched negative curvature. Our main result
combined with a theorem of Mosher, Sageev and Whyte gives QI rigidity
results.
An important ingredient of the proof is a version of the Hilbert–Smith conjecture
for certain metric measure spaces, which uses the full strength of Yang’s theorem on
actions of the p-adic integers on homology manifolds. This might be of independent
interest.
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